Summary: We consider the solution operator \[ (Lu)(t)=A\int^{t}_{0}G(t- \tau)A^{-1}Bu(\tau)d\tau \] corresponding to the abstract equation \(\dot x=Ax+Bu\) on a reflexive Banach space X, where the linear operator \(A: X\supset {\mathcal D}(A)\to X\) is the infinitesimal generator of a (strongly continuous) group G(t) of bounded operators on X, and \(B: U\supset {\mathcal D}(B)\to X\) is a generally unbounded linear operator with \(A^{-1}B\in {\mathcal L}(U,X)\), U being another reflexive Banach space (without loss of generality we take A to be boundedly invertible). Let \(0<T<\infty\) be given.
We prove the following theorem: if L is continuous \(L^ p(0,T;U)\to L^ p(0,T;X)\), \(1<p<\infty\), then in fact L: continuous \(L^ p(0,T;U)\to C([0,T];X)\), a lifting regularity theorem in the time variable. Moreover, we show by a parabolic example with nonhomogeneous term in the Dirichlet boundary conditions that the theorem fails to be true, if G(t) is merely a s.c. semigroup even if holomorphic. Applications of the theorem include mixed hyperbolic problems, including second order scalar hyperbolic equations defined on an open bounded domain \(\Omega \subset {\mathbb{R}}^ n\), \(\partial \Omega =\Gamma\), with nonhomogeneous term of class \(L^ 2(0,T;L^ 2(\Gamma))\) acting in the Dirichlet or in the Neumann boundary conditions. In the former case, the theorem recovers the authors’ original procedure which yielded optimal-regularity results for this dynamics [the authors, Appl. Math. Optim. 10, 275-286 (1983; Zbl 0526.35049)]; in the latter, the theorem improves upon results of J. L. Lions - E. Magenes [Nonhomogeneus boundary value problems and applications, vol. II (1972; Zbl 0227.35001)]. Extension to \(T=\infty\) is also studied.